100BHW5

# 100BHW5 - μ = 0 Derive 2log Λ What is the relationship...

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STAT 100B HWV Problem 1. For Hardy-Weinberg model, suppose the number of AA is 342, the number of Aa is 500, and the number of aa is 187. (1) Calculate the MLE of θ . (2) Test whether the Hardy-Weinberg model fits the data. Calculate 2 log Λ for the likelihood ratio test, and calculate the p -value. Problem 2. Prove that 2 log Λ in Problem 1 can be approximated by χ 2 statistic. Calculate this statistic, and calculate the p -value. Problem 3. Suppose we flip a coin 100 times independently. Let X 1 , X 2 , ..., X 100 be the results of the 100 flips, where X i = 1 if the i -th flip is a head, and X i = 0 otherwise. Suppose we observe 60 heads. We want to test H 0 : p = 1 / 2 versus H 1 : p negationslash = 1 / 2. Calculate 2 log Λ for the likelihood ratio test. Calculate the p -value. Problem 4. Let X 1 , ..., X n N( μ, σ 2 ) independently. (1) Suppose σ 2 is known. We want to test H 0 : μ = μ 0 versus H 1 : μ negationslash = μ 0 , where μ 0 is given (e.g.,
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Unformatted text preview: μ = 0). Derive 2log Λ. What is the relationship between 2log Λ and ( ¯ X − μ ) /σ ? (2) Suppose σ 2 is not known. We want to test the hypotheses in (1), what is 2log Λ? What is the relationship between 2log Λ and ( ¯ X − μ ) /s , where s 2 = ∑ n i =1 ( X i − ¯ X ) 2 / ( n − 1)? (3) Suppose X 1 ,...,X n are-0.4326 -1.6656 0.1253 0.2877 -1.1465 1.1909 1.1892 -0.0376 0.3273 0.1746 We want to test H : μ = 0 versus H 1 : μ n = 0. Compute 2log Λ and compute the p-value. Problem 5. Let X 1 ,...,X n ∼ Exp( λ ). We want to test H : λ = 1 versus H 1 : λ n = 1. (1) Derive 2log Λ. (2) Suppose X 1 ,...,X n are 2.8492 1.0417 0.2068 4.6191 1.9741 1.5957 1.6158 0.5045 1.3013 1.6154 Compute 2log Λ and compute the p-value....
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